in the early part of 1999 it fell back from 35% to 18% as the FTSE index was jumpy but appeared to be bounded between 6300 and 5800.24

Observation of data similar to Figure 2.6, but on the S&P500 index option 3-month volatilities, has motivated Derman (1999) to formulate three different types of market regime:

(a) range-bounded, where future price moves are likely to be constrained within a certain range and there is no significant change in realized volatility;

(b) trending, where the level of the market is changing but in a stable manner so there is again little change in realized volatility in the long run; and

(c) jumpy, where the probability of jumps in the price level is particularly high so realized volatility increases.

Taking a cross-section from Figure 2.6 will normally give a line similar to that shown in Figure 2.7. In Dermans models the skew is approximated by a linear function of the strike. Derman hypothesizes that the form of this linear function should depend on the market regime. Denote by cr(x) the implied volatility of an option with maturity t and strike , ( ) the volatility of the -maturity ATM option, S the current value of the index and 0 and Sa the initial implied volatility and price used to calibrate the tree. Then:

(a) In a range-bounded market skews should be parameterized as

Thus fixed strike volatility oK(x) is independent of the index level in the sense that if the index changes, fixed strike volatilities will not change. This implies that will decrease as the index increases, as can be seen by substituting S - above, giving:

The range-bounded model (2.3a) is called the sticky strike model because local volatilities will be constant with respect to strike. That is, each option has its own binomial tree, with a constant volatility that is determined by the strike of the option. As the index moves all that happens is that the root of the tree is

To understand why this should be so, suppose a firms asset value X and equity price Y vary over time, but in the short run debt is constant. Equity volatility will be greater than asset volatility (Y = X - so YIY = /(X - k), which is greater than AX/X if > 0). In fact equity volatility divided by asset volatility is X/(X - k). Now suppose the equity price jumps down. Then the asset value will go down in the short run, because debt is constant. Then equity volatility will become very much greater than the underlying (but unmeasurable) asset volatility. Therefore a downward jump in equity price should lead to a significant increase in equity volatility in the short run. In an upward-trending market asset values will go up and the equity volatility will go down, becoming closer to the underlying asset volatility. If asset values continue to stay high the asset volatility declines and the firm may wish to take advantage of this. Most likely it will increase its debt level, and if it does, the equity volatility will move up again. The net effect is that equity volatilities can appear to be negatively correlated with price movements, although they remain relatively constant over the longer term.

cfjf(t) = ct0 - b(x)(K - S0).

(2.3a)

atmCO = ct0 - b(x)(S - S0).

(2.4a)

CUCJCJCJCVICJCVICVICVICJCJCJCUCJCJCJCVICVICIJCJCVICJCJCJCJCJCIJCJCJCVI

- \1 - \1 - \1 *1 )<£> - )

Figure 2.7 Volatility skew on 15 March 1999.

moved to the current level of the index. The same tree is still used to price the option.

(b) In a stable trending market skews should be parameterized as

csK(T) = cj0-b(T)(K-S). (2.3b)

Fixed strike volatility ) will increase with the index level but ( ) will be independent of the index, since

ctAtmW = 0. (2.4b)

When the index moves, ATM volatility will remain constant. The trending markets model (2.3b) is called the sticky delta model because local volatilities are constant with respect to the moneyness (or equivalently the delta) of the option. That is to say, it is the moneyness of the option that determines the (still constant) local volatility in the tree. As the index moves, the delta of the option changes and we consequently move to a different tree, the one corresponding to the current option delta.

(c) In jumpy markets skews should be parameterized as

*( ) = <*> - b(x)(K+ S) + 2b(x)S0 (2.3c)

Fixed strike volatility ( ) will decrease when the index goes up, and increase when the index falls. Since

0 = ct0 - 2b(x)(S - S0),

(2.4c)

All three sticky models take the same form when they are expressed in terms of deviations of fixed strike volatility from the ATM volatility

the ATM volatility will also decrease as the index goes up and increase as the index falls, and twice as fast as the fixed strike volatilities do. In the sticky tree model (2.3c) the local volatilities in the binomial tree are no longer constant. There is, however, one unique tree that can be used to price all options, that is determined by the current skew. This is the implied tree described in Derman and Kani (1994).

Alexander (2001a) points out that all three sticky models take the same form when they are expressed in terms of deviations of fixed strike volatility from the ATM volatility. That is,

o*(t) = ctatm(t) = -A(t)(A:-S).

(2.5)

In fact, one can view each of the sticky models as (2.5) together with the alternative parameterization given in equations (2.4a), (2.4b) and (2.4c). Note that each of these equations corresponds to a linear parameterization for the volatility surface a(S, t) in the space dimension. In fact we could specify the deterministic surface

cr(5, t) = b(t)S + c(t),

where the coefficient h(t) jumps between three different levels (0, b and -b) as the market moves between different regimes and the coefficient c(t) represents additional variation in the time dimension.

Not until Chapter 6 shall we develop the tools that allow us to use a more general model, where volatility surfaces are parameterized as a quadratic in S. Therefore, any further discussion of this topic will have to be postponed until §6.3, where Dermans sticky models will be extended to a more flexible framework. For the present let us continue with the linear model framework, and ask how one should determine which value of b(t) should be used. To distinguish which of the sticky models is currently appropriate, a simple approach that is explained in the next subsection is to examine the recent behaviour of ATM volatility and its relationship with the index movements.

2.3.2 Scenario Analysis of Prices and Implied Volatility

A scenario-based risk management of options portfolios requires the definition of scenarios for implied volatilities and underlying asset prices. Given the current smile or skew, what is the appropriate way to change it as the price level moves? In the absence of an effective model of how implied volatilities change with market price, these scenarios may be rather simplistic. Constant volatility scenarios are often augmented with just a few simple scenarios. For example, the 1996 Basle Accord Amendment (§9.1.1) recommends using parallel shifts in all volatilities that are independent of movements in underlying prices.

This section describes a simple method for generating correlated movements between the underlying price and ATM volatility. The discussion above has